Statistical Methods in Quantum Optics 2

Carmichael, H. J.

doi:10.1007/978-3-540-71320-3

Cited by 463 publications

(827 citation statements)

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“…Note also that the terminology of spontaneous emission "rate" only makes sense for a weak or intermediate coupling regime. We stress, however, that the general formalism above rigorously covers both weak and strong coupling regimes in a self-consistent way, naturally recovering phenomena such as vacuum Rabi oscillations and reverse spontaneous emission [40][41][42]. Our expression for the spontaneous emission is also in agreement with derivations from Dung et al [43,44] and Wubs et al [38], who have pioneered many of the general theories of quantized light emission within inhomogeneous materials.…”

Section: Quantum Statistics and Measurement Aspects Of Single Photonssupporting

confidence: 83%

On‐chip single photon sources using planar photonic crystals and single quantum dots

Yao

Rao

Hughes³

2010

Laser & Photonics Reviews

153

184

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We review the basic light-matter interactions and optical properties of chip-based single photon sources, that are enabled by integrating single quantum dots with planar photonic crystals. A theoretical framework is presented that allows one to connect to a wide range of quantum light propagation effects in a physically intuitive and straightforward way. We focus on the important mechanisms of enhanced spontaneous emission, and efficient photon extraction, using all-integrated photonic crystal components including waveguides, cavities, quantum dots and output couplers. The limitations, challenges, and exciting prospects of developing on-chip quantum light sources using integrated photonic crystal structures are discussed.A sequence of optical pulses (top right) interacting with a single quantum dot embedded in a photonic crystal system, resulting in the emission of a train of single photons on-chip.

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Section: Quantum Statistics and Measurement Aspects Of Single Photonssupporting

confidence: 83%

On‐chip single photon sources using planar photonic crystals and single quantum dots

Yao

Rao

Hughes³

2010

Laser & Photonics Reviews

153

184

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“…To complement previous approaches based on the pure-state formalism [15,31], we make a direct connection between the squeezing of the atomic dipole and the squeezing of the cavity field, and point at a distinction between the contributions of singlephoton and two-photon excitation processes.…”

Section: Observation Of Squeezed Light From One Atom Excited With Twomentioning

confidence: 99%

“…The nonlinearity appears at a single-atom and singlephoton level where quantum fluctuations play a major role [5,15] so that its understanding requires a full quantum treatment rather than a simplified linearized approach. In the experiment, the quadrature operator of the light field, X θ , is measured outside the cavity using a homodyne detection with a controllable phase θ.…”

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confidence: 99%

Observation of squeezed light from one atom excited with two photons

Ourjoumtsev

Kubanek

Koch

et al. 2011

Nature

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Single quantum emitters like atoms are wellknown as non-classical light sources which can produce photons one by one at given times [1], with reduced intensity noise. However, the light field emitted by a single atom can exhibit much richer dynamics. A prominent example is the predicted[2] ability for a single atom to produce quadrature-squeezed light [3], with sub-shotnoise amplitude or phase fluctuations. It has long been foreseen, though, that such squeezing would be "at least an order of magnitude more difficult" to observe than the emission of single photons [4]. Squeezed beams have been generated using macroscopic and mesoscopic media down to a few tens of atoms [5], but despite experimental efforts [6][7][8], single-atom squeezing has so far escaped observation. Here we generate squeezed light with a single atom in a highfinesse optical resonator. The strong coupling of the atom to the cavity field induces a genuine quantum mechanical nonlinearity [9], several orders of magnitude larger than for usual macroscopic media [10][11][12]. This produces observable quadrature squeezing [13][14][15] with an excitation beam containing on average only two photons per system lifetime. In sharp contrast to the emission of single photons [16], the squeezed light stems from the quantum coherence of photon pairs emitted from the system [17]. The ability of a single atom to induce strong coherent interactions between propagating photons opens up new perspectives for photonic quantum logic with single emitters [18][19][20][21][22][23].Unlike in a standard Kerr medium, our squeezing does not result from a simple nonlinear polarization of the medium but from a cavity-enhanced atomic coherence which exists for weak coherent driving. Consider a twostate atom with ground and excited states |g and |e . In the absence of a resonator, the amount of squeezing is governed by the atomic coherence, σ = |g e|, and the excited-state occupation probability. The latter pro- * Publication reference: Nature 474, 623-626 (2011), www.nature.com/doifinder/10.1038/nature10170 † Present address: MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands duces incoherent scattering which destroys the squeezing. Therefore, the laser intensity must remain low to preserve the atomic and hence the optical coherence, see Supplementary Information. Under this condition, the optical squeezing is determined by the fluctuations of the atomic coherence, ∆σ 2 = ( σ − σ ) 2 , itself given by ∆σ 2 = − σ 2 owing to the fermionic character of a twostate atom, σ 2 = 0. Note that this sets an upper bound to the amount of squeezing which can be obtained, even when all the light scattered by the atom in all directions is observed.The presence of the cavity introduces two important ingredients as sketched in Fig. 1. First, the cavity mirrors spatially direct the squeezed light towards the detectors thus eliminating the need to observe the full 4π solid angle. Second, the strong coupling to the optical cavity mode makes ...

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“…These two states, |↓ = |0,0 and |↑ = (|1,0 − LETTERS |0,1 )/ √ 2, form an effective two-level system composed of both transmon and cavity degrees of freedom. Within the effective two-level subspace, the photon operators are mapped 22 to Pauli operators a → Σ − / √ 2, a † → Σ + / √ 2, so that the microwave tone acts as a drive on the effective two-level system, that is,…”

mentioning

confidence: 99%

“…a scenario that Carmichael and co-workers 12,22 have referred to as 'dressing of dressed states'. The Hamiltonian H eff refers to the frame rotating at the drive frequency, ∆ = ω 01 − g 0 − ω d is the detuning between the drive and the vacuum Rabi peak and Ω = √ 2ξ is the effective drive strength.…”

mentioning

confidence: 99%

Nonlinear response of the vacuum Rabi resonance

et al. 2008

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On the level of single atoms and photons, the coupling between atoms and the electromagnetic field is typically very weak. By using a cavity to confine the field, the strength of this interaction can be increased by many orders of magnitude, to a point where it dominates over any dissipative process. This strong-coupling regime of cavity quantum electrodynamics 1,2 has been reached for real atoms in optical cavities 3 , and for artificial atoms in circuit quantum electrodynamics 4 and quantum dot systems 5,6 . A signature of strong coupling is the splitting of the cavity transmission peak into a pair of resolvable peaks when a single resonant atom is placed inside the cavity, an effect known as vacuum Rabi splitting. The circuit quantum electrodynamics architecture is ideally suited for going beyond this linear-response effect. Here, we show that increasing the drive power results in two unique nonlinear features in the transmitted heterodyne signal: the supersplitting of each vacuum Rabi peak into a doublet and the appearance of extra peaks with the characteristic √ n spacing of the Jaynes-Cummings ladder. These findings constitute direct evidence for the coupling between the quantized microwave field and the anharmonic spectrum of a superconducting qubit acting as an artificial atom.Circuit quantum electrodynamics (QED) realizes the coupling between a superconducting qubit and the microwave field inside an on-chip transmission-line resonator 4,7 . The drastic reduction in mode volume for such a quasi-one-dimensional system 8,9 , the recent improvement in coherence times 10 and the absence of atomic motion render circuit QED an ideal system for studying the strong-coupling limit. In the customary linear-response measurement of the transmitted microwave radiation, the vacuum Rabi splitting manifests itself as an avoided crossing between the qubit and the resonator.It may be argued that the observation of the splitting is not yet a clear sign for quantum behaviour 11,12 , because avoided crossings are also ubiquitous in classical physics. However, quantum mechanics gives rise to a distinct anharmonicity of these splittings when initializing the resonator in a higher photon Fock state: when the resonator mode is occupied by n photons, the splitting is enhanced by a factor √ n + 1 as compared with the vacuum Rabi situation. This anharmonicity has been observed with single atoms in both microwave 13 and optical cavities 14,15 . In circuit QED, this characteristic trait has been observed in time-domain measurements 16 . In a system similar to ours, the position of the n = 2 peaks was recently demonstrated in a two-tone pump-probe measurement 17 .Here, we present a theoretical analysis and experimental investigation of the vacuum Rabi resonance in a circuit QED system, where we study the √ n anharmonicity up to n = 5 by exploring the power dependence of the heterodyne transmission (see the Methods section). This type of detection is particularly simple, because it is a continuous measurement involving only a single dr...

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Statistical Methods in Quantum Optics 2

Cited by 463 publications

References 0 publications

On‐chip single photon sources using planar photonic crystals and single quantum dots

On‐chip single photon sources using planar photonic crystals and single quantum dots

Observation of squeezed light from one atom excited with two photons

Nonlinear response of the vacuum Rabi resonance

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